Will he finish in time? No way! Yes! Strong Jump-Traceability The Computably Enumerable Case Peter Cholak University of Notre Dame Department of Mathematics www.nd.edu/~cholak Supported by NSF Grant DMS 02-45167 (USA). Logic Colloquium 07 Preprint available: Cholak, Downey, and Greenberg, Strong Jump-Traceability I: the Computably Enumerable Case .
Will he finish in time? No way! Yes! Reals with little value as oracles Are there any? How low do they go? Are they all the same? Try to understand the relation between reals with low initial segment complexity as measured by Kolmogorov complexity and reals with low computational power (as measured by the halting set relative to the real). Example: Loveland showed the a real α is computable iff the sequence C(α ↾ n) − C(n) is bounded, where C is plain Kolmogorov complexity.
Will he finish in time? No way! Yes! K -Trivial Reals Reals with very low initial segment complexity Definition If the sequence K(A ↾ n) − K(n) is bounded then A is K -trivial , where K is prefix-free Kolmogorov complexity. Theorem (Chatin, Downey, Hirschfeldt, Nies , Solovay, Stephan) The K -trivial reals form a robust nontrivial ideal of low ∆ 0 2 degrees.
Will he finish in time? No way! Yes! Cost Functions How to build an K -trivial real. Or how do you prove your results. Definition The cost (or weight) of x at stage s is � 2 − K s (n) . c(x, s) = x<n<s � Example: Define a computably enumerable set A = s A s by putting x ∈ A s + 1 − A s if W e,s ∩ A s = ∅ , x > 2 e , x ∈ W e,s and c(x, s) < 2 − (e + 1 ) . Then A is simple and K -trivial.
Will he finish in time? No way! Yes! C.e. Traceability Computationally Feeble Definition • A (c.e.) trace is an uniformly c.e. sequence � T x � of finite sets. (Equivalently there is a computable function g such that for all x , T x = W g(x) .) • A trace traces a function f if for all x , f(x) ∈ T x . • A function h : ω → ω \ { 0 } is an order if h is computable, nondecreasing and lim s h(s) = ∞ . • The tracing obeys an order h if for all x , | T x | ≤ h(x) . • A degree a is c.e. traceable if there is an order h such that every f ≤ T a can be traced by some trace obeying h . Theorem (Zambella) If A is K -trivial then deg (A) is c.e. traceable.
Will he finish in time? No way! Yes! Jump Traceable More Computationally Feeble Definition A is jump-traceable if there is some order h and a c.e. trace � T x � obeying h and tracing { e } X (e) (if { e } X (e) ↓ ) then { e } X (e) ∈ T e ). Theorem (Nies) Jump-traceability and superlowness are the same on the c.e. sets. There are non K -trivial jump traceable sets. Theorem (Nies, Figueira, and Stephan) If A is K -trivial, then A is jump traceable with respect to an order roughly h(n) = n log n .
Will he finish in time? No way! Yes! Strongly Jump Traceable Even More Computationally Feeble Definition A is strongly jump-traceable iff { e } X (e) can be traced obeying any order. Theorem (Nies, Figueira, and Stephan) There are non-computable, strongly jump-traceable, computably enumerable reals. Strong jump-traceability is weaker than jump-traceability on the c.e. reals. Question (Nies and Miller) Is the class of K -trivials exactly the class of strongly jump traceable reals? Is strongly jump traceability a combinatorial characterization of K -triviality?
Will he finish in time? No way! Yes! N0! The c.e. strongly jump-traceable degrees form a proper subideal of the K -trivials. Theorem Every c.e. strongly jump-traceable set is K -trivial. Theorem There is a K -trivial c.e. set that is not strongly jump-traceable. Indeed it is not jump traceable with a bound of size roughly log log n . Theorem The c.e. strongly jump-traceable degrees form an ideal. Corollary (to the proof of the first theorem above) � If a set A is jump-traceable with respect to about log n then it is K -trivial.
Will he finish in time? No way! Yes! An hierarchy of jump-traceability? Or a possible combinatorial characterization of the K -trivials. � log n < n log n . Question n ∈ N 2 − h(n) < ∞ , A is Is A K -trivial iff for all orders h with � jump traceable with order h ?
Recommend
More recommend
